Rumor Arrays

Rumor sequences are generated recursively as follows: fix nonnegative
integers b, c, and let z0 = 0. For
n ≥ 1, define zn =
(bzn-1
+ c) mod n, where the least nonnegative residue modulo
n is taken. There have been a few papers dealing with the
behavior of rumor sequences, but they all concern that behavior when
the value of c is fixed. It turns out that if, for a given value
of b, the rumor sequences for c = 0,1,2,... are written
down, one below the other, some interesting and unexpected patterns
appear in the columns of that array. These patterns are investigated,
proving some, and, based on computer generated data, we make two
conjectures.